Question

given $de \parallel ac$
prove $\frac{ab}{db} = \frac{cb}{eb}$

(image of triangle abc with de parallel to ac, d on ab, e on cb)

statement | reason
--- | ---
$de \parallel ac$ | given
$\square$ | when parallel lines are cut by a transversal, corresponding angles are congruent.
$\square$ | reflexive property
$\square$ | aa similarity
$\square$ | corresponding sides of similar triangles are proportional.

options:
$\because \frac{ab}{db} = \frac{cb}{eb}$
$\because \frac{ab}{db} = \frac{ce}{eb}$
$\because \angle b \cong \angle b$
$\because \angle c \cong \angle c$
$\because \angle d \cong \angle a$
$\because \angle a \cong \angle e$
$\because \triangle abc \sim \triangle dbe$

Explanation:

Step1: Identify Corresponding Angles

Since \( DE \parallel AC \), when transversal \( AB \) cuts them, \( \angle D \cong \angle A \) (corresponding angles). When transversal \( CB \) cuts them, \( \angle E \cong \angle C \) (though we'll use the first pair and the common angle).

Step2: Identify Common Angle

\( \angle B \cong \angle B \) (reflexive property).

Step3: Apply AA Similarity

With \( \angle D \cong \angle A \) and \( \angle B \cong \angle B \), by AA (Angle - Angle) similarity, \( \triangle ABC \sim \triangle DBE \).

Step4: Use Proportional Sides

Since \( \triangle ABC \sim \triangle DBE \), corresponding sides are proportional. So, \( \frac{AB}{DB}=\frac{CB}{EB} \) (corresponding sides \( AB \) & \( DB \), \( CB \) & \( EB \)).

Now, filling the table:

• First blank (corresponding angles): \( \boldsymbol{\angle D \cong \angle A} \)
• Second blank (reflexive property): \( \boldsymbol{\angle B \cong \angle B} \)
• Third blank (AA Similarity): \( \boldsymbol{\triangle ABC \sim \triangle DBE} \)
• Fourth blank (proportional sides): \( \boldsymbol{\frac{AB}{DB}=\frac{CB}{EB}} \)

Answer:

• First row (Reason: When parallel lines...): \( \angle D \cong \angle A \)
• Second row (Reason: Reflexive property): \( \angle B \cong \angle B \)
• Third row (Reason: AA Similarity): \( \triangle ABC \sim \triangle DBE \)
• Fourth row (Reason: Corresponding sides...): \( \frac{AB}{DB}=\frac{CB}{EB} \)